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diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index bc892ca9..e0be9ed3 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.v @@ -1,3 +1,4 @@ +(* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) @@ -6,155 +7,453 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ChoiceFacts.v 8132 2006-03-05 10:59:47Z herbelin $ i*) +(*i $Id: ChoiceFacts.v 8892 2006-06-04 17:59:53Z herbelin $ i*) -(** We show that the functional formulation of the axiom of Choice - (usual formulation in type theory) is equivalent to its relational - formulation (only formulation of set theory) + the axiom of - (parametric) definite description (aka axiom of unique choice) *) +(** ** Some facts and definitions concerning choice and description in + intuitionistic logic. -(** This shows that the axiom of choice can be assumed (under its - relational formulation) without known inconsistency with classical logic, - though definite description conflicts with classical logic *) +We investigate the relations between the following choice and +description principles + +- AC_rel = relational form of the (non extensional) axiom of choice + (a "set-theoretic" axiom of choice) +- AC_fun = functional form of the (non extensional) axiom of choice + (a "type-theoretic" axiom of choice) +- AC! = functional relation reification + (known as axiom of unique choice in topos theory, + sometimes called principle of definite description in + the context of constructive type theory) + +- GAC_rel = guarded relational form of the (non extensional) axiom of choice +- GAC_fun = guarded functional form of the (non extensional) axiom of choice +- GAC! = guarded functional relation reification + +- OAC_rel = "omniscient" relational form of the (non extensional) axiom of choice +- OAC_fun = "omniscient" functional form of the (non extensional) axiom of choice + (called AC* in Bell [Bell]) +- OAC! + +- ID_iota = intuitionistic definite description +- ID_epsilon = intuitionistic indefinite description + +- D_iota = (weakly classical) definite description principle +- D_epsilon = (weakly classical) indefinite description principle + +- PI = proof irrelevance +- IGP = independence of general premises + (an unconstrained generalisation of the constructive principle of + independence of premises) +- Drinker = drinker's paradox (small form) + (called Ex in Bell [Bell]) + +We let also + +IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal predicate logic +IPL_2 = 2nd-order impredicative minimal predicate logic +IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.) + +Table of contents + +A. Definitions + +B. IPL_2^2 |- AC_rel + AC! = AC_fun + +C. 1. AC_rel + PI -> GAC_rel and PL_2 |- AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel + +C. 2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker + +D. Derivability of choice for decidable relations with well-ordered codomain + +E. Equivalence of choices on dependent or non dependent functional types + +F. Non contradiction of constructive descriptions wrt functional choices + +G. Definite description transports classical logic to the computational world -Section ChoiceEquivalences. +References: + +[Bell] John L. Bell, Choice principles in intuitionistic set theory, +unpublished. + +[Bell93] John L. Bell, Hilbert's Epsilon Operator in Intuitionistic +Type Theories, Mathematical Logic Quarterly, volume 39, 1993. + +[Carlstrøm05] Jesper Carlstrøm, Interpreting descriptions in +intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005. +*) + +Set Implicit Arguments. + +Notation Local "'inhabited' A" := A (at level 10, only parsing). + +(**********************************************************************) +(** *** A. Definitions *) + +(** Choice, reification and description schemes *) + +Section ChoiceSchemes. Variables A B :Type. -Definition RelationalChoice := - forall (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y) -> - exists R' : A -> B -> Prop, - (forall x:A, - exists y : B, R x y /\ R' x y /\ (forall y':B, R' x y' -> y = y')). +Variables P:A->Prop. + +Variables R:A->B->Prop. + +(** **** Constructive choice and description *) + +(** AC_rel *) + +Definition RelationalChoice_on := + forall R:A->B->Prop, + (forall x : A, exists y : B, R x y) -> + (exists R' : A->B->Prop, subrelation R' R /\ forall x, exists! y, R' x y). + +(** AC_fun *) + +Definition FunctionalChoice_on := + forall R:A->B->Prop, + (forall x : A, exists y : B, R x y) -> + (exists f : A->B, forall x : A, R x (f x)). + +(** AC! or Functional Relation Reification (known as Axiom of Unique Choice + in topos theory; also called principle of definite description *) + +Definition FunctionalRelReification_on := + forall R:A->B->Prop, + (forall x : A, exists! y : B, R x y) -> + (exists f : A->B, forall x : A, R x (f x)). + +(** ID_epsilon (constructive version of indefinite description; + combined with proof-irrelevance, it may be connected to + Carlstrøm's type theory with a constructive indefinite description + operator) *) + +Definition ConstructiveIndefiniteDescription_on := + forall P:A->Prop, + (exists x, P x) -> { x:A | P x }. + +(** ID_iota (constructive version of definite description; combined + with proof-irrelevance, it may be connected to Carlstrøm's and + Stenlund's type theory with a constructive definite description + operator) *) + +Definition ConstructiveDefiniteDescription_on := + forall P:A->Prop, + (exists! x, P x) -> { x:A | P x }. + +(** **** Weakly classical choice and description *) -Definition FunctionalChoice := - forall (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y) -> - exists f : A -> B, (forall x:A, R x (f x)). +(** GAC_rel *) -Definition ParamDefiniteDescription := - forall (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y /\ (forall y':B, R x y' -> y = y')) -> - exists f : A -> B, (forall x:A, R x (f x)). +Definition GuardedRelationalChoice_on := + forall P : A->Prop, forall R : A->B->Prop, + (forall x : A, P x -> exists y : B, R x y) -> + (exists R' : A->B->Prop, + subrelation R' R /\ forall x, P x -> exists! y, R' x y). + +(** GAC_fun *) + +Definition GuardedFunctionalChoice_on := + forall P : A->Prop, forall R : A->B->Prop, + inhabited B -> + (forall x : A, P x -> exists y : B, R x y) -> + (exists f : A->B, forall x, P x -> R x (f x)). + +(** GFR_fun *) + +Definition GuardedFunctionalRelReification_on := + forall P : A->Prop, forall R : A->B->Prop, + inhabited B -> + (forall x : A, P x -> exists! y : B, R x y) -> + (exists f : A->B, forall x : A, P x -> R x (f x)). + +(** OAC_rel *) + +Definition OmniscientRelationalChoice_on := + forall R : A->B->Prop, + exists R' : A->B->Prop, + subrelation R' R /\ forall x : A, (exists y : B, R x y) -> exists! y, R' x y. + +(** OAC_fun *) + +Definition OmniscientFunctionalChoice_on := + forall R : A->B->Prop, + inhabited B -> + exists f : A->B, forall x : A, (exists y : B, R x y) -> R x (f x). + +(** D_epsilon *) + +Definition ClassicalIndefiniteDescription := + forall P:A->Prop, + A -> { x:A | (exists x, P x) -> P x }. + +(** D_iota *) + +Definition ClassicalDefiniteDescription := + forall P:A->Prop, + A -> { x:A | (exists! x, P x) -> P x }. + +End ChoiceSchemes. + +(** Generalized schemes *) + +Notation RelationalChoice := + (forall A B, RelationalChoice_on A B). +Notation FunctionalChoice := + (forall A B, FunctionalChoice_on A B). +Notation FunctionalChoiceOnInhabitedSet := + (forall A B, inhabited B -> FunctionalChoice_on A B). +Notation FunctionalRelReification := + (forall A B, FunctionalRelReification_on A B). + +Notation GuardedRelationalChoice := + (forall A B, GuardedRelationalChoice_on A B). +Notation GuardedFunctionalChoice := + (forall A B, GuardedFunctionalChoice_on A B). +Notation GuardedFunctionalRelReification := + (forall A B, GuardedFunctionalRelReification_on A B). + +Notation OmniscientRelationalChoice := + (forall A B, OmniscientRelationalChoice_on A B). +Notation OmniscientFunctionalChoice := + (forall A B, OmniscientFunctionalChoice_on A B). + +Notation ConstructiveDefiniteDescription := + (forall A, ConstructiveDefiniteDescription_on A). +Notation ConstructiveIndefiniteDescription := + (forall A, ConstructiveIndefiniteDescription_on A). + +(** Subclassical schemes *) + +Definition ProofIrrelevance := + forall (A:Prop) (a1 a2:A), a1 = a2. + +Definition IndependenceOfGeneralPremises := + forall (A:Type) (P:A -> Prop) (Q:Prop), + inhabited A -> + (Q -> exists x, P x) -> exists x, Q -> P x. + +Definition SmallDrinker'sParadox := + forall (A:Type) (P:A -> Prop), inhabited A -> + exists x, (exists x, P x) -> P x. + +(**********************************************************************) +(** *** B. AC_rel + PDP = AC_fun + + We show that the functional formulation of the axiom of Choice + (usual formulation in type theory) is equivalent to its relational + formulation (only formulation of set theory) + the axiom of + (parametric) definite description (aka axiom of unique choice) *) + +(** This shows that the axiom of choice can be assumed (under its + relational formulation) without known inconsistency with classical logic, + though definite description conflicts with classical logic *) Lemma description_rel_choice_imp_funct_choice : - ParamDefiniteDescription -> RelationalChoice -> FunctionalChoice. -intros Descr RelCh. -red in |- *; intros R H. -destruct (RelCh R H) as [R' H0]. -destruct (Descr R') as [f H1]. -intro x. -elim (H0 x); intros y [H2 [H3 H4]]; exists y; split; [ exact H3 | exact H4 ]. + forall A B : Type, + FunctionalRelReification_on A B -> RelationalChoice_on A B -> FunctionalChoice_on A B. +Proof. +intros A B Descr RelCh R H. +destruct (RelCh R H) as (R',(HR'R,H0)). +destruct (Descr R') as (f,Hf). +firstorder. exists f; intro x. -elim (H0 x); intros y [H2 [H3 H4]]. -rewrite <- (H4 (f x) (H1 x)). -exact H2. +destruct (H0 x) as (y,(HR'xy,Huniq)). +rewrite <- (Huniq (f x) (Hf x)). +apply HR'R; assumption. Qed. -Lemma funct_choice_imp_rel_choice : FunctionalChoice -> RelationalChoice. -intros FunCh. -red in |- *; intros R H. -destruct (FunCh R H) as [f H0]. -exists (fun x y => y = f x). -intro x; exists (f x); split; - [ apply H0 - | split; [ reflexivity | intros y H1; symmetry in |- *; exact H1 ] ]. +Lemma funct_choice_imp_rel_choice : + forall A B, FunctionalChoice_on A B -> RelationalChoice_on A B. +Proof. +intros A B FunCh R H. +destruct (FunCh R H) as (f,H0). +exists (fun x y => f x = y). +split. + intros x y Heq; rewrite <- Heq; trivial. + intro x; exists (f x); split. + reflexivity. + trivial. Qed. -Lemma funct_choice_imp_description : - FunctionalChoice -> ParamDefiniteDescription. -intros FunCh. -red in |- *; intros R H. +Lemma funct_choice_imp_description : + forall A B, FunctionalChoice_on A B -> FunctionalRelReification_on A B. +Proof. +intros A B FunCh R H. destruct (FunCh R) as [f H0]. (* 1 *) intro x. -elim (H x); intros y [H0 H1]. -exists y; exact H0. +destruct (H x) as (y,(HRxy,_)). +exists y; exact HRxy. (* 2 *) exists f; exact H0. Qed. Theorem FunChoice_Equiv_RelChoice_and_ParamDefinDescr : - FunctionalChoice <-> RelationalChoice /\ ParamDefiniteDescription. -split. + forall A B, FunctionalChoice_on A B <-> + RelationalChoice_on A B /\ FunctionalRelReification_on A B. +Proof. +intros A B; split. intro H; split; [ exact (funct_choice_imp_rel_choice H) | exact (funct_choice_imp_description H) ]. intros [H H0]; exact (description_rel_choice_imp_funct_choice H0 H). Qed. -End ChoiceEquivalences. +(**********************************************************************) +(** *** C. Connection between the guarded, non guarded and descriptive choices and *) (** We show that the guarded relational formulation of the axiom of Choice comes from the non guarded formulation in presence either of the independance of premises or proof-irrelevance *) -Definition GuardedRelationalChoice (A B:Type) := - forall (P:A -> Prop) (R:A -> B -> Prop), - (forall x:A, P x -> exists y : B, R x y) -> - exists R' : A -> B -> Prop, - (forall x:A, - P x -> - exists y : B, R x y /\ R' x y /\ (forall y':B, R' x y' -> y = y')). - -Definition ProofIrrelevance := forall (A:Prop) (a1 a2:A), a1 = a2. +(**********************************************************************) +(** **** C. 1. AC_rel + PI -> GAC_rel and AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel *) Lemma rel_choice_and_proof_irrel_imp_guarded_rel_choice : - (forall A B, RelationalChoice A B) - -> ProofIrrelevance -> (forall A B, GuardedRelationalChoice A B). + RelationalChoice -> ProofIrrelevance -> GuardedRelationalChoice. Proof. intros rel_choice proof_irrel. red in |- *; intros A B P R H. -destruct (rel_choice _ _ (fun (x:sigT P) (y:B) => R (projT1 x) y)) as [R' H0]. -intros [x HPx]. -destruct (H x HPx) as [y HRxy]. +destruct (rel_choice _ _ (fun (x:sigT P) (y:B) => R (projT1 x) y)) as (R',(HR'R,H0)). +intros (x,HPx). +destruct (H x HPx) as (y,HRxy). exists y; exact HRxy. set (R'' := fun (x:A) (y:B) => exists H : P x, R' (existT P x H) y). -exists R''; intros x HPx. -destruct (H0 (existT P x HPx)) as [y [HRxy [HR'xy Huniq]]]. -exists y. split. - exact HRxy. - split. - red in |- *; exists HPx; exact HR'xy. - intros y' HR''xy'. +exists R''; split. + intros x y (HPx,HR'xy). + change x with (projT1 (existT P x HPx)); apply HR'R; exact HR'xy. + intros x HPx. + destruct (H0 (existT P x HPx)) as (y,(HR'xy,Huniq)). + exists y; split. exists HPx; exact HR'xy. + intros y' (H'Px,HR'xy'). apply Huniq. - unfold R'' in HR''xy'. - destruct HR''xy' as [H'Px HR'xy']. - rewrite proof_irrel with (a1 := HPx) (a2 := H'Px). - exact HR'xy'. + rewrite proof_irrel with (a1 := HPx) (a2 := H'Px); exact HR'xy'. Qed. -Definition IndependenceOfGeneralPremises := - forall (A:Type) (P:A -> Prop) (Q:Prop), - (Q -> exists x, P x) -> exists x, Q -> P x. - Lemma rel_choice_indep_of_general_premises_imp_guarded_rel_choice : - forall A B, RelationalChoice A B -> - IndependenceOfGeneralPremises -> GuardedRelationalChoice A B. -Proof. -intros A B RelCh IndPrem. -red in |- *; intros P R H. -destruct (RelCh (fun x y => P x -> R x y)) as [R' H0]. - intro x. apply IndPrem. - apply H. - exists R'. - intros x HPx. - destruct (H0 x) as [y [H1 H2]]. - exists y. split. - apply (H1 HPx). - exact H2. + forall A B, inhabited B -> RelationalChoice_on A B -> + IndependenceOfGeneralPremises -> GuardedRelationalChoice_on A B. +Proof. +intros A B Inh AC_rel IndPrem P R H. +destruct (AC_rel (fun x y => P x -> R x y)) as (R',(HR'R,H0)). + intro x. apply IndPrem. exact Inh. intro Hx. + apply H; assumption. + exists (fun x y => P x /\ R' x y). + firstorder. +Qed. + +Lemma guarded_rel_choice_imp_rel_choice : + forall A B, GuardedRelationalChoice_on A B -> RelationalChoice_on A B. +Proof. +intros A B GAC_rel R H. +destruct (GAC_rel (fun _ => True) R) as (R',(HR'R,H0)). + firstorder. +exists R'; firstorder. Qed. +(** OAC_rel = GAC_rel *) + +Lemma guarded_iff_omniscient_rel_choice : + GuardedRelationalChoice <-> OmniscientRelationalChoice. +Proof. +split. + intros GAC_rel A B R. + apply (GAC_rel A B (fun x => exists y, R x y) R); auto. + intros OAC_rel A B P R H. + destruct (OAC_rel A B R) as (f,Hf); exists f; firstorder. +Qed. + +(**********************************************************************) +(** **** C. 2. AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker *) + +(** AC_fun + IGP = GAC_fun *) + +Lemma guarded_fun_choice_imp_indep_of_general_premises : + GuardedFunctionalChoice -> IndependenceOfGeneralPremises. +Proof. +intros GAC_fun A P Q Inh H. +destruct (GAC_fun unit A (fun _ => Q) (fun _ => P) Inh) as (f,Hf). +tauto. +exists (f tt); auto. +Qed. + +Lemma guarded_fun_choice_imp_fun_choice : + GuardedFunctionalChoice -> FunctionalChoiceOnInhabitedSet. +Proof. +intros GAC_fun A B Inh R H. +destruct (GAC_fun A B (fun _ => True) R Inh) as (f,Hf). +firstorder. +exists f; auto. +Qed. + +Lemma fun_choice_and_indep_general_prem_imp_guarded_fun_choice : + FunctionalChoiceOnInhabitedSet -> IndependenceOfGeneralPremises + -> GuardedFunctionalChoice. +Proof. +intros AC_fun IndPrem A B P R Inh H. +apply (AC_fun A B Inh (fun x y => P x -> R x y)). +intro x; apply IndPrem; eauto. +Qed. + +(** AC_fun + Drinker = OAC_fun *) + +(** This was already observed by Bell [Bell] *) + +Lemma omniscient_fun_choice_imp_small_drinker : + OmniscientFunctionalChoice -> SmallDrinker'sParadox. +Proof. +intros OAC_fun A P Inh. +destruct (OAC_fun unit A (fun _ => P)) as (f,Hf). +auto. +exists (f tt); firstorder. +Qed. + +Lemma omniscient_fun_choice_imp_fun_choice : + OmniscientFunctionalChoice -> FunctionalChoiceOnInhabitedSet. +Proof. +intros OAC_fun A B Inh R H. +destruct (OAC_fun A B R Inh) as (f,Hf). +exists f; firstorder. +Qed. + +Lemma fun_choice_and_small_drinker_imp_omniscient_fun_choice : + FunctionalChoiceOnInhabitedSet -> SmallDrinker'sParadox + -> OmniscientFunctionalChoice. +Proof. +intros AC_fun Drinker A B R Inh. +destruct (AC_fun A B Inh (fun x y => (exists y, R x y) -> R x y)) as (f,Hf). + intro x; apply (Drinker B (R x) Inh). + exists f; assumption. +Qed. + +(** OAC_fun = GAC_fun *) + +(** This is derivable from the intuitionistic equivalence between IGP and Drinker +but we give a direct proof *) + +Lemma guarded_iff_omniscient_fun_choice : + GuardedFunctionalChoice <-> OmniscientFunctionalChoice. +Proof. +split. + intros GAC_fun A B R Inh. + apply (GAC_fun A B (fun x => exists y, R x y) R); auto. + intros OAC_fun A B P R Inh H. + destruct (OAC_fun A B R Inh) as (f,Hf). + exists f; firstorder. +Qed. + +(**********************************************************************) +(** *** D. Derivability of choice for decidable relations with well-ordered codomain *) (** Countable codomains, such as [nat], can be equipped with a well-order, which implies the existence of a least element on inhabited decidable subsets. As a consequence, the relational form of the axiom of choice is derivable on [nat] for decidable relations. - We show instead that definite description and the functional form - of the axiom of choice are equivalent on decidable relation with [nat] - as codomain + We show instead that functional relation reification and the + functional form of the axiom of choice are equivalent on decidable + relation with [nat] as codomain *) Require Import Wf_nat. @@ -163,12 +462,11 @@ Require Import Decidable. Require Import Arith. Definition has_unique_least_element (A:Type) (R:A->A->Prop) (P:A->Prop) := - (exists x, (P x /\ forall x', P x' -> R x x') - /\ forall x', P x' /\ (forall x'', P x'' -> R x' x'') -> x=x'). + exists! x, P x /\ forall x', P x' -> R x x'. Lemma dec_inh_nat_subset_has_unique_least_element : forall P:nat->Prop, (forall n, P n \/ ~ P n) -> - (exists n, P n) -> has_unique_least_element nat le P. + (exists n, P n) -> has_unique_least_element le P. Proof. intros P Pdec (n0,HPn0). assert @@ -194,30 +492,228 @@ assert assumption. destruct H0. rewrite Heqn; assumption. -destruct (H n0) as [(n,(Hltn,(Hmin,Huniqn)))|]; [exists n | exists n0]; - repeat split; + destruct (H n0) as [(n,(Hltn,(Hmin,Huniqn)))|]; [exists n | exists n0]; + repeat split; assumption || intros n' (HPn',Hminn'); apply le_antisym; auto. Qed. -Definition FunctionalChoice_on (A B:Type) (R:A->B->Prop) := - (forall x:A, exists y : B, R x y) -> - exists f : A -> B, (forall x:A, R x (f x)). +Definition FunctionalChoice_on_rel (A B:Type) (R:A->B->Prop) := + (forall x:A, exists y : B, R x y) -> + exists f : A -> B, (forall x:A, R x (f x)). Lemma classical_denumerable_description_imp_fun_choice : forall A:Type, - ParamDefiniteDescription A nat -> - forall R, (forall x y, decidable (R x y)) -> FunctionalChoice_on A nat R. + FunctionalRelReification_on A nat -> + forall R:A->nat->Prop, + (forall x y, decidable (R x y)) -> FunctionalChoice_on_rel R. Proof. intros A Descr. red in |- *; intros R Rdec H. set (R':= fun x y => R x y /\ forall y', R x y' -> y <= y'). -destruct (Descr R') as [f Hf]. +destruct (Descr R') as (f,Hf). intro x. apply (dec_inh_nat_subset_has_unique_least_element (R x)). apply Rdec. apply (H x). exists f. intros x. -destruct (Hf x) as [Hfx _]. +destruct (Hf x) as (Hfx,_). +assumption. +Qed. + +(**********************************************************************) +(** *** E. Choice on dependent and non dependent function types are equivalent *) + +(** **** E. 1. Choice on dependent and non dependent function types are equivalent *) + +Definition DependentFunctionalChoice_on (A:Type) (B:A -> Type) := + forall R:forall x:A, B x -> Prop, + (forall x:A, exists y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +Notation DependentFunctionalChoice := + (forall A (B:A->Type), DependentFunctionalChoice_on B). + +(** The easy part *) + +Theorem dep_non_dep_functional_choice : + DependentFunctionalChoice -> FunctionalChoice. +Proof. +intros AC_depfun A B R H. + destruct (AC_depfun A (fun _ => B) R H) as (f,Hf). + exists f; trivial. +Qed. + +(** Deriving choice on product types requires some computation on + singleton propositional types, so we need computational + conjunction projections and dependent elimination of conjunction + and equality *) + +Scheme and_indd := Induction for and Sort Prop. +Scheme eq_indd := Induction for eq Sort Prop. + +Definition proj1_inf (A B:Prop) (p : A/\B) := + let (a,b) := p in a. + +Theorem non_dep_dep_functional_choice : + FunctionalChoice -> DependentFunctionalChoice. +Proof. +intros AC_fun A B R H. +pose (B' := { x:A & B x }). +pose (R' := fun (x:A) (y:B') => projT1 y = x /\ R (projT1 y) (projT2 y)). +destruct (AC_fun A B' R') as (f,Hf). +intros x. destruct (H x) as (y,Hy). +exists (existT (fun x => B x) x y). split; trivial. +exists (fun x => eq_rect _ _ (projT2 (f x)) _ (proj1_inf (Hf x))). +intro x; destruct (Hf x) as (Heq,HR) using and_indd. +destruct (f x); simpl in *. +destruct Heq using eq_indd; trivial. +Qed. + +(** **** E. 2. Reification of dependent and non dependent functional relation are equivalent *) + +Definition DependentFunctionalRelReification_on (A:Type) (B:A -> Type) := + forall (R:forall x:A, B x -> Prop), + (forall x:A, exists! y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +Notation DependentFunctionalRelReification := + (forall A (B:A->Type), DependentFunctionalRelReification_on B). + +(** The easy part *) + +Theorem dep_non_dep_functional_rel_reification : + DependentFunctionalRelReification -> FunctionalRelReification. +Proof. +intros DepFunReify A B R H. + destruct (DepFunReify A (fun _ => B) R H) as (f,Hf). + exists f; trivial. +Qed. + +(** Deriving choice on product types requires some computation on + singleton propositional types, so we need computational + conjunction projections and dependent elimination of conjunction + and equality *) + +Theorem non_dep_dep_functional_rel_reification : + FunctionalRelReification -> DependentFunctionalRelReification. +Proof. +intros AC_fun A B R H. +pose (B' := { x:A & B x }). +pose (R' := fun (x:A) (y:B') => projT1 y = x /\ R (projT1 y) (projT2 y)). +destruct (AC_fun A B' R') as (f,Hf). +intros x. destruct (H x) as (y,(Hy,Huni)). + exists (existT (fun x => B x) x y). repeat split; trivial. + intros (x',y') (Heqx',Hy'). + simpl in *. + destruct Heqx'. + rewrite (Huni y'); trivial. +exists (fun x => eq_rect _ _ (projT2 (f x)) _ (proj1_inf (Hf x))). +intro x; destruct (Hf x) as (Heq,HR) using and_indd. +destruct (f x); simpl in *. +destruct Heq using eq_indd; trivial. +Qed. + +(**********************************************************************) +(** *** F. Non contradiction of constructive descriptions wrt functional axioms of choice *) + +(** **** F. 1. Non contradiction of indefinite description *) + +Lemma relative_non_contradiction_of_indefinite_desc : + (ConstructiveIndefiniteDescription -> False) + -> (FunctionalChoice -> False). +Proof. +intros H AC_fun. +assert (AC_depfun := non_dep_dep_functional_choice AC_fun). +pose (A0 := { A:Type & { P:A->Prop & exists x, P x }}). +pose (B0 := fun x:A0 => projT1 x). +pose (R0 := fun x:A0 => fun y:B0 x => projT1 (projT2 x) y). +pose (H0 := fun x:A0 => projT2 (projT2 x)). +destruct (AC_depfun A0 B0 R0 H0) as (f, Hf). +apply H. +intros A P H'. +exists (f (existT (fun _ => sigT _) A + (existT (fun P => exists x, P x) P H'))). +pose (Hf' := + Hf (existT (fun _ => sigT _) A + (existT (fun P => exists x, P x) P H'))). assumption. Qed. + +Lemma constructive_indefinite_descr_fun_choice : + ConstructiveIndefiniteDescription -> FunctionalChoice. +Proof. +intros IndefDescr A B R H. +exists (fun x => proj1_sig (IndefDescr B (R x) (H x))). +intro x. +apply (proj2_sig (IndefDescr B (R x) (H x))). +Qed. + +(** **** F. 2. Non contradiction of definite description *) + +Lemma relative_non_contradiction_of_definite_descr : + (ConstructiveDefiniteDescription -> False) + -> (FunctionalRelReification -> False). +Proof. +intros H FunReify. +assert (DepFunReify := non_dep_dep_functional_rel_reification FunReify). +pose (A0 := { A:Type & { P:A->Prop & exists! x, P x }}). +pose (B0 := fun x:A0 => projT1 x). +pose (R0 := fun x:A0 => fun y:B0 x => projT1 (projT2 x) y). +pose (H0 := fun x:A0 => projT2 (projT2 x)). +destruct (DepFunReify A0 B0 R0 H0) as (f, Hf). +apply H. +intros A P H'. +exists (f (existT (fun _ => sigT _) A + (existT (fun P => exists! x, P x) P H'))). +pose (Hf' := + Hf (existT (fun _ => sigT _) A + (existT (fun P => exists! x, P x) P H'))). +assumption. +Qed. + +Lemma constructive_definite_descr_fun_reification : + ConstructiveDefiniteDescription -> FunctionalRelReification. +Proof. +intros DefDescr A B R H. +exists (fun x => proj1_sig (DefDescr B (R x) (H x))). +intro x. +apply (proj2_sig (DefDescr B (R x) (H x))). +Qed. + +(**********************************************************************) +(** *** G. Excluded-middle + definite description => computational excluded-middle *) + +(** The idea for the following proof comes from [ChicliPottierSimpson02] *) + +(** Classical logic and axiom of unique choice (i.e. functional + relation reification), as shown in [ChicliPottierSimpson02], + implies the double-negation of excluded-middle in [Set] (which is + incompatible with the impredicativity of [Set]). + + We adapt the proof to show that constructive definite description + transports excluded-middle from [Prop] to [Set]. + + [ChicliPottierSimpson02] Laurent Chicli, Loïc Pottier, Carlos + Simpson, Mathematical Quotients and Quotient Types in Coq, + Proceedings of TYPES 2002, Lecture Notes in Computer Science 2646, + Springer Verlag. *) + +Require Import Setoid. + +Theorem constructive_definite_descr_excluded_middle : + ConstructiveDefiniteDescription -> + (forall P:Prop, P \/ ~ P) -> (forall P:Prop, {P} + {~ P}). +Proof. +intros Descr EM P. +pose (select := fun b:bool => if b then P else ~P). +assert { b:bool | select b } as ([|],HP). + apply Descr. + rewrite <- unique_existence; split. + destruct (EM P). + exists true; trivial. + exists false; trivial. + intros [|] [|] H1 H2; simpl in *; reflexivity || contradiction. +left; trivial. +right; trivial. +Qed. |